65 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
65 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
66 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
66 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
67 just erasing the blob from the picture |
67 just erasing the blob from the picture |
68 (but keeping the blob label $u$). |
68 (but keeping the blob label $u$). |
69 |
69 |
70 \nn{it seems rather strange to make this a theorem} |
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71 \nn{it's a theorem because it's stated in the introduction, and I wanted everything there to have numbers that pointed into the paper --S} |
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72 Note that directly from the definition we have |
70 Note that directly from the definition we have |
73 \begin{thm} |
71 \begin{prop} |
74 \label{thm:skein-modules} |
72 \label{thm:skein-modules} |
75 The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
73 The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
76 \end{thm} |
74 \end{prop} |
77 This also establishes the second |
75 This also establishes the second |
78 half of Property \ref{property:contractibility}. |
76 half of Property \ref{property:contractibility}. |
79 |
77 |
80 Next, we want the vector space $\bc_2(X)$ to capture ``the space of all relations |
78 Next, we want the vector space $\bc_2(X)$ to capture ``the space of all relations |
81 (redundancies, syzygies) among the |
79 (redundancies, syzygies) among the |
290 twig label $u_i$ as having the form |
288 twig label $u_i$ as having the form |
291 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
289 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
292 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
290 and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
293 |
291 |
294 For lack of a better name, |
292 For lack of a better name, |
295 \nn{can we think of a better name?} |
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296 we'll call elements of $P$ cone-product polyhedra, |
293 we'll call elements of $P$ cone-product polyhedra, |
297 and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set). |
294 and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set). |
298 \end{remark} |
295 \end{remark} |
299 |
296 |